Dear Dr. SETI:I need to know how to calculate Effective Isotropic Radiated Power for a communications link. Let's say I have ten Watts into an omnidirectional antenna. Isn't the EIRP just 10 Watts (+40 dBi)? I assume that since the antenna is omnidirectional, it has unity gain -- right?

Jack, Mojave Desert

The Doctor Responds:

It's not quite so simple, Jack. First, let's define an isotropic antenna. It is one that has absolutely uniform gain in all directions in spherical space (that is, it illuminates all 4 pi steradians of space equally). It is the only truly (3 dimensional) omnidirectional antenna, in that it radiates equally poorly in all directions.

Unfortunately, the isotropic antenna is a fiction. You can not build one, buy one, or find one in nature. So, it is a mathematical model, nothing more.

The Sun comes close to isotropic radiation (as does any other star), but it is not exactly isotropic, because the sun itself is not perfectly spherical. Since it is a spinning gas ball, it bulges at the equator, and is thinner across the poles. It is an oblate spheroid (kind of like what happens to all of us as we age!) If it were perfectly spherical (that is, not spinning or bulging), then its radiation pattern would be isotropic.

Now, for multipoint telecommunications purposes, an omnidirectional antenna assumes we're living back in 1491, when the Earth was still flat. It radiates equally poorly in all directions in two-dimensional space. A vertically oriented dipole does this pretty well. But, its pattern is not really isotropic, because it doesn't radiate off the ends of the dipole (that is, straight up or straight down, perpendicular to plane of the assumed flat Earth).

Compared to an isotrope, a dipole has some gain. That is, because of conservation of energy, all the radiation has to be accounted for. The energy that would otherwise have come off the ends has to go somewhere, so it increases the signal level coming off the antenna broadside to the dipole. This little bit of extra radiation can be accounted for by quantifying the gain of the diploe, in dB relative to what the radiation would have been if the antenna had been truly isotropic, rather than being a dipole. The unit of measure for the dipole's gain is then dBi (deciBels compared to an isotrope), and for a perfect dipole, can be shown (mathematically) to equal +2.15 dBi.

For what it's worth, that means that, compared to an isotrope, a dipole produces a voltage gain of 1.28, or a power gain of (1.28)^2 = 1.64.

Let's say you stack two vertically oriented dipoles, one above the other (vertically), and feed them in the proper phase. We've now built a two-element collinear omnidirectional array. There's still a sharp null off the ends of the dipoles (that is, straight up and straight down). But, all the radiation still has to be accounted for (remember conservation of energy?) The radiation pattern, in the plane of the (assumed flat) Earth, is now stronger, so the gain (relative to our mythical isotrope) has increased. In an ideal world, doubling the number of dipoles (from 1 to 2) increases the gain by 3 dB, so the array how has a gain of +5.15 dBi. (Your mileage may vary, of course, because the cables used to phase the two dipoles are not lossless, and the dipoles are not perfect.) Go to four stacked dipoles, and we can (theoretically) add another 3 dB, and so forth.

All these multi-element collinears can be omnidirectional (or, at least, could have been, back in 1491), but they are hardly isotropic. This is why you can't really calculate EIRP unless you know something about the actual antenna being used. Something other than simply "it's omnidirectional."

With all this as background, let's define EIRP (effective isotropic radiated power). It is simply the power in a given direction, at a given distance, measured from a specific transmitter into a specific antenna, compared to the power you would read from exactly the same transmitter, at the same distance and in the same direction, if it were coupled to an isotropic antenna. Mathematically, you compute EIRP by adding the logarithmic expression of transmitter power (typically in dBm or dBw -- deciBels compared to a Watt, or deciBels compared to a milliWatt) to the logarithmic expression of its antenna's gain (measured in dBi -- deciBels relative to isotropic radiation).

Notice that I am adding antenna gain to transmitter power. This is because they are both expressed in logarithmic units. If I knew them both in linear units (that is, transmitter power in Watts, and antenna gain as a power ratio), I would of course multiply the two to get EIRP (which would then be in Watts relative to isotropic, as opposed to the logarithmic unit, deciBels relative to a Watt into an isotrope, dBWi, or deciBels relative to a milliwatt into an isotrope -- dBmi).

So, that's how we calculate EIRP, Jack -- even in the Mojave Desert (which, I understand, is about as close as we can get to living back in 1491).